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<li><a class="reference internal" href="#">Empirical evaluation of the impact of k-means initialization</a></li>
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<div class="sphx-glr-example-title section" id="empirical-evaluation-of-the-impact-of-k-means-initialization">
<span id="sphx-glr-auto-examples-cluster-plot-kmeans-stability-low-dim-dense-py"></span><h1>Empirical evaluation of the impact of k-means initialization<a class="headerlink" href="#empirical-evaluation-of-the-impact-of-k-means-initialization" title="Permalink to this headline">¶</a></h1>
<p>Evaluate the ability of k-means initializations strategies to make
the algorithm convergence robust as measured by the relative standard
deviation of the inertia of the clustering (i.e. the sum of squared
distances to the nearest cluster center).</p>
<p>The first plot shows the best inertia reached for each combination
of the model (<code class="docutils literal notranslate"><span class="pre">KMeans</span></code> or <code class="docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code>) and the init method
(<code class="docutils literal notranslate"><span class="pre">init=&quot;random&quot;</span></code> or <code class="docutils literal notranslate"><span class="pre">init=&quot;kmeans++&quot;</span></code>) for increasing values of the
<code class="docutils literal notranslate"><span class="pre">n_init</span></code> parameter that controls the number of initializations.</p>
<p>The second plot demonstrate one single run of the <code class="docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code>
estimator using a <code class="docutils literal notranslate"><span class="pre">init=&quot;random&quot;</span></code> and <code class="docutils literal notranslate"><span class="pre">n_init=1</span></code>. This run leads to
a bad convergence (local optimum) with estimated centers stuck
between ground truth clusters.</p>
<p>The dataset used for evaluation is a 2D grid of isotropic Gaussian
clusters widely spaced.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Author: Olivier Grisel &lt;olivier.grisel@ensta.org&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">matplotlib.cm</span> <span class="k">as</span> <span class="nn">cm</span>

<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">shuffle</span>
<span class="kn">from</span> <span class="nn">sklearn.utils</span> <span class="kn">import</span> <span class="n">check_random_state</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">MiniBatchKMeans</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">KMeans</span>

<span class="n">random_state</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>

<span class="c1"># Number of run (with randomly generated dataset) for each strategy so as</span>
<span class="c1"># to be able to compute an estimate of the standard deviation</span>
<span class="n">n_runs</span> <span class="o">=</span> <span class="mi">5</span>

<span class="c1"># k-means models can do several random inits so as to be able to trade</span>
<span class="c1"># CPU time for convergence robustness</span>
<span class="n">n_init_range</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">20</span><span class="p">])</span>

<span class="c1"># Datasets generation parameters</span>
<span class="n">n_samples_per_center</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">grid_size</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">scale</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="n">n_clusters</span> <span class="o">=</span> <span class="n">grid_size</span> <span class="o">**</span> <span class="mi">2</span>


<span class="k">def</span> <span class="nf">make_data</span><span class="p">(</span><span class="n">random_state</span><span class="p">,</span> <span class="n">n_samples_per_center</span><span class="p">,</span> <span class="n">grid_size</span><span class="p">,</span> <span class="n">scale</span><span class="p">):</span>
    <span class="n">random_state</span> <span class="o">=</span> <span class="n">check_random_state</span><span class="p">(</span><span class="n">random_state</span><span class="p">)</span>
    <span class="n">centers</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="n">i</span><span class="p">,</span> <span class="n">j</span><span class="p">]</span>
                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">)</span>
                        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">grid_size</span><span class="p">)])</span>
    <span class="n">n_clusters_true</span><span class="p">,</span> <span class="n">n_features</span> <span class="o">=</span> <span class="n">centers</span><span class="o">.</span><span class="n">shape</span>

    <span class="n">noise</span> <span class="o">=</span> <span class="n">random_state</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span>
        <span class="n">scale</span><span class="o">=</span><span class="n">scale</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_samples_per_center</span><span class="p">,</span> <span class="n">centers</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">1</span><span class="p">]))</span>

    <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([</span><span class="n">c</span> <span class="o">+</span> <span class="n">noise</span> <span class="k">for</span> <span class="n">c</span> <span class="ow">in</span> <span class="n">centers</span><span class="p">])</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">concatenate</span><span class="p">([[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">n_samples_per_center</span>
                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters_true</span><span class="p">)])</span>
    <span class="k">return</span> <span class="n">shuffle</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">)</span>

<span class="c1"># Part 1: Quantitative evaluation of various init methods</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plots</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">legends</span> <span class="o">=</span> <span class="p">[]</span>

<span class="n">cases</span> <span class="o">=</span> <span class="p">[</span>
    <span class="p">(</span><span class="n">KMeans</span><span class="p">,</span> <span class="s1">&#39;k-means++&#39;</span><span class="p">,</span> <span class="p">{}),</span>
    <span class="p">(</span><span class="n">KMeans</span><span class="p">,</span> <span class="s1">&#39;random&#39;</span><span class="p">,</span> <span class="p">{}),</span>
    <span class="p">(</span><span class="n">MiniBatchKMeans</span><span class="p">,</span> <span class="s1">&#39;k-means++&#39;</span><span class="p">,</span> <span class="p">{</span><span class="s1">&#39;max_no_improvement&#39;</span><span class="p">:</span> <span class="mi">3</span><span class="p">}),</span>
    <span class="p">(</span><span class="n">MiniBatchKMeans</span><span class="p">,</span> <span class="s1">&#39;random&#39;</span><span class="p">,</span> <span class="p">{</span><span class="s1">&#39;max_no_improvement&#39;</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="s1">&#39;init_size&#39;</span><span class="p">:</span> <span class="mi">500</span><span class="p">}),</span>
<span class="p">]</span>

<span class="k">for</span> <span class="n">factory</span><span class="p">,</span> <span class="n">init</span><span class="p">,</span> <span class="n">params</span> <span class="ow">in</span> <span class="n">cases</span><span class="p">:</span>
    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Evaluation of </span><span class="si">%s</span><span class="s2"> with </span><span class="si">%s</span><span class="s2"> init&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">factory</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">init</span><span class="p">))</span>
    <span class="n">inertia</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">empty</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">n_init_range</span><span class="p">),</span> <span class="n">n_runs</span><span class="p">))</span>

    <span class="k">for</span> <span class="n">run_id</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_runs</span><span class="p">):</span>
        <span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_data</span><span class="p">(</span><span class="n">run_id</span><span class="p">,</span> <span class="n">n_samples_per_center</span><span class="p">,</span> <span class="n">grid_size</span><span class="p">,</span> <span class="n">scale</span><span class="p">)</span>
        <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">n_init</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">n_init_range</span><span class="p">):</span>
            <span class="n">km</span> <span class="o">=</span> <span class="n">factory</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="n">n_clusters</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="n">init</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="n">run_id</span><span class="p">,</span>
                         <span class="n">n_init</span><span class="o">=</span><span class="n">n_init</span><span class="p">,</span> <span class="o">**</span><span class="n">params</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
            <span class="n">inertia</span><span class="p">[</span><span class="n">i</span><span class="p">,</span> <span class="n">run_id</span><span class="p">]</span> <span class="o">=</span> <span class="n">km</span><span class="o">.</span><span class="n">inertia_</span>
    <span class="n">p</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">errorbar</span><span class="p">(</span><span class="n">n_init_range</span><span class="p">,</span> <span class="n">inertia</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">inertia</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span>
    <span class="n">plots</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
    <span class="n">legends</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">%s</span><span class="s2"> with </span><span class="si">%s</span><span class="s2"> init&quot;</span> <span class="o">%</span> <span class="p">(</span><span class="n">factory</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">init</span><span class="p">))</span>

<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;n_init&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;inertia&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">plots</span><span class="p">,</span> <span class="n">legends</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Mean inertia for various k-means init across </span><span class="si">%d</span><span class="s2"> runs&quot;</span> <span class="o">%</span> <span class="n">n_runs</span><span class="p">)</span>

<span class="c1"># Part 2: Qualitative visual inspection of the convergence</span>

<span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">make_data</span><span class="p">(</span><span class="n">random_state</span><span class="p">,</span> <span class="n">n_samples_per_center</span><span class="p">,</span> <span class="n">grid_size</span><span class="p">,</span> <span class="n">scale</span><span class="p">)</span>
<span class="n">km</span> <span class="o">=</span> <span class="n">MiniBatchKMeans</span><span class="p">(</span><span class="n">n_clusters</span><span class="o">=</span><span class="n">n_clusters</span><span class="p">,</span> <span class="n">init</span><span class="o">=</span><span class="s1">&#39;random&#39;</span><span class="p">,</span> <span class="n">n_init</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
                     <span class="n">random_state</span><span class="o">=</span><span class="n">random_state</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_clusters</span><span class="p">):</span>
    <span class="n">my_members</span> <span class="o">=</span> <span class="n">km</span><span class="o">.</span><span class="n">labels_</span> <span class="o">==</span> <span class="n">k</span>
    <span class="n">color</span> <span class="o">=</span> <span class="n">cm</span><span class="o">.</span><span class="n">nipy_spectral</span><span class="p">(</span><span class="nb">float</span><span class="p">(</span><span class="n">k</span><span class="p">)</span> <span class="o">/</span> <span class="n">n_clusters</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">[</span><span class="n">my_members</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[</span><span class="n">my_members</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;o&#39;</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;.&#39;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">color</span><span class="p">)</span>
    <span class="n">cluster_center</span> <span class="o">=</span> <span class="n">km</span><span class="o">.</span><span class="n">cluster_centers_</span><span class="p">[</span><span class="n">k</span><span class="p">]</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">cluster_center</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">cluster_center</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="s1">&#39;o&#39;</span><span class="p">,</span>
             <span class="n">markerfacecolor</span><span class="o">=</span><span class="n">color</span><span class="p">,</span> <span class="n">markeredgecolor</span><span class="o">=</span><span class="s1">&#39;k&#39;</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Example cluster allocation with a single random init</span><span class="se">\n</span><span class="s2">&quot;</span>
              <span class="s2">&quot;with MiniBatchKMeans&quot;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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